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MARKOFF–ROSENBERGER TRIPLES IN ARITHMETIC PROGRESSION 3 1 ENRIQUEGONZA´LEZ–JIMEŃEZANDJOSE´ M.TORNERO 0 2 Abstract. WestudythesolutionsoftheRosenberg–Markoffequationax2+ n by2+cz2=dxyz (ageneralization ofthe well–knownMarkoffequation). We a specificallyfocus onlookingforsolutions inarithmeticprogressionthat liein J the ring of integers of a number field. With the help of previous work by 1 Alvanos and Poulakis, we give a complete decision algorithm, which allows 2 us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: ] thegeneralizedMarkoffequationx2+y2+z2=dxyzoverquadraticfieldsand T the classic Markoff equation x2+y2+z2 = 3xyz over an arbitrary number N field. . h t a m 1. Variations on the Markoff equation [ The Markoff equation is the Diophantine equation 1 v x2+y2+z2 =3xyz; x,y,z Z ; + 9 ∈ 2 which was studied first by Markoff in [12, 13]. In those papers, many interesting 0 properties related to the solutions of this equation were proved. Among other 5 things, Markoff showed there were infinitely many solutions (so–called Markoff . 1 triples), gave a procedure to construct new solutions from old ones and proved 0 that, in fact, all integral solutions could be constructed from one fundamental so- 3 lution (1,1,1). 1 Since then, the Markoff equation and its solutions have been object of intense : v research. Remarkably,Frobenius, while studying the MarkoffequationoverGauss- i X ian integers [8], noticed that, for a given ordered solution x y z, there was no ≤ ≤ other ordered solution x′ y′ z. This conjecture, widely known as the Frobe- r a nius unicity conjecture, ha≤s rem≤ained open since, although some important partial results have been settled [2, 3, 5, 17, 6]. Our work started with the Markoffequation, but from a different point of view. Rather than looking at the Frobenius conjecture, we decided to focus on looking for Markofftriples with some extra structure: more precisely,those Markofftriples which are in arithmetic progression (a.p. in what follows). However, in our study, itbecameapparentthatageneralizationcouldbeconsideredinsteadoftheoriginal Markoff equation. From the many extended versions of the Markoff equation, two have been by far the most studied in the literature. 2010 Mathematics Subject Classification. Primary: 11D25;Secondary: 11D45,14G05. Key words and phrases. Markoffequation, arithmeticprogression. The first author was partially supported by the grant MTM2009–07291. The second author waspartiallysupportedbythegrantsFQM–218andP08–FQM–03894, FSEandFEDER(EU). 1 2 ENRIQUEGONZA´LEZ–JIMEŃEZANDJOSE´ M.TORNERO First, the Hurwitz (or Markoff–Hurwitz) equation [9], given by x2+x2+...+x2 =ax x ...x , 1 2 n 1 2 n forwhichHurwitzhimselfprovedthatallsolutionscouldbe constructedfromaset of easy mappings acting on a finite set of particular solutions. The other succesfully studied generalization mentioned above, and the one who will center much of our work, is the so–called Markoff–Rosenbergerequation [16]: ax2+by2+cz2 =dxyz. Note that, in [16] (and in all subsequent articles concerning the Rosenberger generalization) it is further required that ad, bd, cd. This comes from Rosen- | | | berger’s originalmotivation, related to binary forms and Fuchsian groups. We will not assume this for most of our work. As far as we know, there are no results in the literature concerning the Markoff–Rosenbergerequation with no conditions. Let us consider then the equation ax2+by2+cz2 =dxyz; and assume we have a solution in a.p. which may be written in the form x=X, y =X +Y, z =X+2Y. The equation then becomes dX3+3dX2Y +2dXY2 (a+b+c)X2 (2b+4c)XY (b+4c)Y2 =0, − − − which is the equation of a cubic curve. For what follows we will be taking homogeneous coordinates [X : Y : Z] and considering a projective closure of our curve: :dX3+3dX2Y +2dXY2 (a+b+c)X2Z (2b+4c)XYZ (b+4c)Y2Z =0, C − − − which has, regardless of (a,b,c,d), a singular point (actually a node) at [0 : 0 : 1] and three points at infinity: [0 : 1 : 0], [1 : 1 : 0] and [2 : 1 : 0]. Our aim was − − finding all integral affine points of . C This situation makes particularly simple the use of the INTEGRAL-POINTSalgo- rithmby Alvanos andPoulakis[1]. Whatmakesthis algorithmmoreremarkableis thefactthatitworksforarbitrarynumberfieldscomputingaffinepointswhoseco- ordinateslie inthe correspondingringofintegers. So,fromMarkoffintegraltriples we had come to Markoff–Rosenberger triples which lie in the ring of integers of a number field (always in a.p.). In the third section we will develop an algorithm to compute all the Markoff– Rosenberger triples over a number field K for fixed a,b,c,d , where K K ∈ O O denotes the ring of integers of K. In particular, this algorithm allows us to give a characterization of the values a,b,c,d such that there exists a non–trivial triple in a.p. over (see Proposition 1). In the fourth section we show several finite- K O ness results related to our problem. In the last section we will deal first with the generalized Markoff equation, obtaining theoretical results over the rationals and overimaginaryquadraticfields. Moreover,wewillincludesomeextensivecomputa- tions we have performed. These computations lead to a well–supported conjecture overquadratic fields that might encourage future researchin this area. Finally, we will work with the classic Markoff equation but over arbitrary number fields with bounded discriminant. MARKOFF–ROSENBERGER TRIPLES IN ARITHMETIC PROGRESSION 3 Butfirst,astheINTEGRAL-POINTSalgorithmwillbeheavilyusedinwhatfollows, we will recall its steps, for the convenience of the reader. The proofs concerning correctnessandtermination,aswellasmanyotherinterestingfeaturescanbefound in the original reference [1]. 2. An algorithmic short trip Alvanos and Poulakis developed in [1] a very polished algorithm for the compu- tation of the set of affine points on a genus zero curve whose coordinates can be chosen to lie in a ring of algebraic integers. The version presented here is actually the so–called(by the authors)INTEGRAL-POINTS3,where 3 refersto the number of points at infinity. The precise algorithm goes as follows: fix a number field K and we are given a curve, say , defined by an affine equation C F(X,Y)=0, where F(X,Y) K[X,Y], deg(F)=deg (F)=N; ∈ Y verifying that has exactly three smooth points at infinity V ,V ,V . We want 1 2 3 C { } to compute the affine points of ( ). K C O Step 1. Compute the singular points of and save those in ( ). K C C O Step 2. Find number fields M , M such that K M K and polynomials 1 2 i ⊂ ⊂ a M [X,Y], b M [X]; i i i i ∈ ∈ such that deg (a )<N and Y i f (X,Y):=a (X,Y)/b (X) (V V ), i i i 3 i ∈L − where (V V ) denotes the Riemann-Roch space of the divisor V V . 3 i 3 i L − − Step 3. Compute α ,β suchthat α f andβ /f are integralover [X]. i i ∈OMi i i i i OMi This step is carried out by an algorithm called DENOMINATORS which is presented before in [1]. Step 4. Determine maximal sets A of elements which are not pairwise i ⊂ OMi associate and such that its norm divides that of α β . i i Step 5. Let M be the normal closure of the composition of M and M . Solve 1 2 then, in M, the equation c f +c f =1. 1 1 2 2 Step 6. For every (k ,k ) A A determine the (finite) solution set S(k ,k ) 1 2 1 2 1 2 ∈ × of the unit equation c k c k 1 1 2 2 U + U =1. 1 2 (cid:18) α (cid:19) (cid:18) α (cid:19) 1 2 The finiteness ofthis setofsolutionsis a well-knownfactwhichgoesbackto Siegel [18]. A more recent account on how to actually compute this set of solutions can be found, for instance in [20] or the most efficient algorithm of Wildanger [21]. Step 7. For any (k ,k ) A A and (u ,u ) S(k ,k ) compute the resultant 1 2 1 2 1 2 1 2 ∈ × ∈ R (X)=Res (F(X,Y),α a (X,Y) k u b (X)), (k1,u1) Y 1 1 − 1 1 1 and determine, the set S of solutions in for some R . OK (k1,u1) Step 8. For any v S compute the possible pairs (v,w) ( ). K ∈ ∈C O Step 9. The affine points of ( ) are those computed in Step 1 and those K C O computed in Step 8. 4 ENRIQUEGONZA´LEZ–JIMEŃEZANDJOSE´ M.TORNERO 3. Integral points on curves at work First we must put our curve in a suitable form for INTEGRAL-POINTS3, being carefulinordertopreserveintegralpoints. Thiscanbeachievedmakingthechange X Y X, Y X, Z Z; 7−→ − 7−→ 7−→ which preserves not only integral projective points but also points at infinity (this is important regarding the algorithm), hence obtaining :dY3 (a+b+c)Y2Z dX2Y +2(a c)XYZ (a+c)X2Z =0. D − − − − Our new curve has still the same singular point [0 : 0 : 1] (a node) and three points at infinity: P =[1:1:0], P =[1: 1:0], P =[1:0:0], 1 2 3 − all of them smooth. Our aim is then to find all affine points in ( ). K D O MovingontoStep2wemustfindthengeneratorsfor (P P )and (P P ). 3 1 3 2 L − L − These are given by easy calculation (a´ la Riemann–Roch, so to say): dY2 (a+b+c)Y dX2+(a 3c)X f (X,Y)= − − − (P P ) 1 3 1 (a+c)X ∈L − dY2+(a+b+c)Y +dX2+( 3a+c)X f (X,Y)= − − (P P ) 2 3 2 (a+c)X ∈L − Next (Step 3), consider α =α =a+c, β =b+4c, β =b+4a, 1 2 1 2 and define R (T)=T3+(dX +a+5c)T2+(a+c)(2dX +b+8c)T +(a+c)2(b+4c), 1 R (T)=T3+( dX +5a+c)T2+(a+c)( 2dX+8a+b)T +(a+c)2(b+4a), 2 − − S (T)=T3+(2dX +b+8c)T2+(b+4c)(dX +a+5c)T +(a+c)(b+4c)2, 1 S (T)=T3+( 2dX +8a+b)T2+(4a+b)( dX +5a+c)T +(a+c)(b+4a)2, 2 − − for which β i R (α f (X,Y))=S =0, i=1,2. i i i i (cid:18)f (X,Y)(cid:19) i Now we consider (Step 4) the following sets: (3.1) A = k (k ) divides (α β ) / , i i K K i K i i { ∈O |N N } ∼ where denotes we are actually interested in the equivalence class (modulo as- ∼ sociated elements) and denotes the absolute norm map. Note that this step K N depends on K. As for Step 5 is concerned, we have f +f = 2. 1 2 − Now we must consider (again depending on K), for every (k ,k ) (A ,A ), 1 2 1 2 ∈ the finite set S of solutions to the unit equation k1,k2 (3.2) k u +k u = 2(a+c). 1 1 2 2 − In Step 7, for any (k ,k ) (A ,A ) and for any (u ,u ) S , we have 1 2 ∈ 1 2 1 2 ∈ k1,k2 R (X) = Res (F(X,Y),(a+c)a (X,Y) k u b (X)) k1,u1 Y 1 − 1 1 1 = (a+c)3d2X3(X z ), − − k1,u1 where (a+c+k u )(4ck u +k2u2+(a+c)(b+4c)) z = 1 1 1 1 1 1 . k1,u1 u u k k d 1 2 1 2 MARKOFF–ROSENBERGER TRIPLES IN ARITHMETIC PROGRESSION 5 As the X–coordinates of all affine points in ( ) appear as roots (in , of K K D O O course) of some R (X), we have two candidates X =0 and X =z . k1,u1 k1,u1 Next step is computing the possible points. For X =0 we have a+b+c 0, (K). (cid:18) d (cid:19)∈D For X =z we have k1,u1 a+c z , z (K), (cid:18) k1,u1 a+c+k u k1,u1(cid:19)∈D 1 1 and, remarkably, (a+c)(b+4c)+(2a+b+2c)k u √∆ (a+c+k u ) 1 1 1 1 zk1,u1,h 2u u k k d± i ∈D(L), 1 2 1 2 −   where L=K √∆ and (cid:16) (cid:17) ∆ = 4(k u )4+8(a+3c)(k u )3+(4a2+b2+8ab+56ac+8bc+52c2)(k u )2 1 1 1 1 1 1 +2(a+c)(b+4c)(6a+b+6c)k u +(a+c)2(b+4c)2 1 1 Hence, we have the following characterization. Proposition 1. Let K be a number field and a,b,c,d . Then, there exists a K ∈O non–trivial solution in a.p. over for the Markoff–Rosenberger equation ax2+ K O by2+cz2 =dxyz if and only if one of the following conditions hold: (a) d (a+b+c). | (b) d (a+c+k u )(k2u2+4ck u +(a+c)(b+4c)), for some(k ,k ) (A ,A ) | 1 1 1 1 1 1 1 2 ∈ 1 2 and some (u ,u ) S . 1 2 ∈ k1,k2 4. Finiteness results We will write, for a given ring R, R/R∗ for the set of elements of R up to multiplication by a unit (as customary), and R/R2 for the set of elements of R with no square root in R. Let us call from now on, for a given number field K and given a,b,c,d , K ∈O (K)= –solutions in a.p. to ax2+by2+cz2 =dxyz , (a,b,c,d) K AP O (cid:8) (cid:9) where obviously we always have the trivial solution (0,0,0). Remember that we are disregarding the Rosenberger conditions ad, bd, cd. | | | First,mindthatwehaveset–upabijectionbetween (K)andtheaffine (a,b,c,d) AP points of ( ), which we already know to be a finite set [11, 10] (because has K D O D three points at infinity). Theorem 2. Let K be a number field and a,b,c,d . Then (K) is K (a,b,c,d) ∈ O AP finite. The followingresults follow directly from the characterizationgivenonProposi- tion 1 above and deep results of Corvaja and Zannier [7]: Theorem 3. Let K be a number field and a,b,c . Then K ∈O (1) We have # d / ∗ (K)= (0,0,0) < . ∈OK OK |AP(a,b,c,d) 6 { } ∞ (cid:8) (cid:9) 6 ENRIQUEGONZA´LEZ–JIMEŃEZANDJOSE´ M.TORNERO (2) If ∆ / 2 , then ∈OK OK # d / ∗ (K)( K √∆ < . n ∈OK OK |AP(a,b,c,d) AP(a,b,c,d)(cid:16) (cid:16) (cid:17)(cid:17)o ∞ (3) Let L/K be a finite algebraic extension. Then # d / ∗ (K)( (L) < . ∈OK OK |AP(a,b,c,d) AP(a,b,c,d) ∞ (cid:8) (cid:9) (4) We have # (d,∆) / ∗ / 2 (K)( K √∆ < . n ∈OK OK ×OK OK |AP(a,b,c,d) AP(a,b,c,d)(cid:16) (cid:16) (cid:17)(cid:17)o ∞ (5) Let D Z and denote by (K)the set of algebraic extensions of degree >0 D ∈ A D of K up to isomorphism. Then # (d,L) / ∗ (K) (K)( (L) < . ∈OK OK ×AD | AP(a,b,c,d) AP(a,b,c,d) ∞ (cid:8) (cid:9) Moreover, we have #  (L)< . (a,b,c,d) AP ∞ L∈A[D(K)d∈O[K/OK∗  Proof. The first three statements come directly from Proposition 1. As for the remaining cases, let us recall Corollary 1 from [7] for the case which stated K O that if is a plane curve with three or more points at infinity, K a number field C and D Z then if L runs through all algebraic extension of degree D of K: >0 ∈ # ( L)< . C O ∞ [L:K[]≤D   Applying this above result to , we get the desired statements. (cid:3) D Remark 1. Rosenberger proved that, with the extra conditions we mentioned in the first section, the only equations with non–trivial integral solutions were those given by (a,b,c,d) (1,1,1,1), (1,1,1,3), (1,1,2,2), (1,1,2,4), (1,2,3,6), (1,1,5,5) . ∈{ } All of them have solutions in a.p. The first five of them verify condition (a) on the characterizationgiven at Proposition 1. The last one x2+y2+5z2 =5xyz, doesnot,butitverifiesthesecondcondition(twosolutionina.p. being( 3, 1,1) − − and ( 7, 1,5)). − − 5. Computational results We have implemented in Magma [4] the algorithm developed on section 3. Note that,givenanumberfieldthereareonlythreemajorproblemstosolve: tocompute thesetsA givenon(3.1),tosolvetheunitequationgivenby(3.2)andtodetermine i if an algebraic number is integral; they are sorted out by the Magma functions NormEquation,UnitEquationand IsIntegral,respectively. Our original goal, the study of Markoff triples in a.p. can by now be easily achieved. MARKOFF–ROSENBERGER TRIPLES IN ARITHMETIC PROGRESSION 7 Theorem 4. (Q)= (0,0,0),(3,3,3),( 15, 6,3),(3, 6, 15) . (1,1,1,1) • AP { − − − − } (Q)= (0,0,0),(1,1,1),( 5, 2,1),(1, 2, 5) . (1,1,1,3) • AP { − − − − } If d=1,3, then (Q)= (0,0,0) . (1,1,1,d) • 6 AP { } Actually, Markoff himself proved [12, 13] that, if d = 1,3, the equation has no 6 integer solutions, so the general case is trivial. For the cases d = 1,3 we run our algorithm to obtain the above results. Note that a simpler algorithm is available thanks to Poulakis and Voskos [15] when the base field is Q. So, as a direct application of the algorithms explained above, we tried next to study exhaustively the generalized Markoff equation x2+y2+z2 =dxyz, d Z ∈ looking for solution in a.p. over arbitrary quadratic fields. When we move to this, more general, case, we find a groundbreaking paper by Silverman[19]. Inthatpaper,knownfactsfromtheintegralcase(howtoobtainall solutions, number of points of bounded height and so on) are carefully generalized for the imaginary quadratic case. For this case we obtain the following result. Theorem 5. Let D Z/Z2, D <0 and i=√ 1. Then ∈ − (Q(i)) = (0,0,0),(3,3,3),( 15, 6,3),(3, 6, 15), (1,1,1,1) • AP { − − − − (2, 2i+2, 4i+2),( i+2,2, i+2), ± ± ± ∓ ( i+2, 2i 1, 3i 4),( 2i 1, 1, 2i 1), ± ± − ± − ± − − ∓ − ( 4i+2,2i+2,2),( 3i 4, 2i 1, i+2) ± ± − ± − ± } (Q(i)) = (0,0,0),( 2i+1, i+1,1),(1, i+1, 2i+1) . (1,1,1,2) • AP { ± ± ± ± } (Q(√D)) = (Q), if (D,d)=( 1,1),( 1,2). (1,1,1,d) (1,1,1,d) • AP AP 6 − − Proof. Letus put a=b=c=1 in the algorithm. Under these hypothesis, the sets A given on (3.1) satisfy A = A and, if k A then it must hold (k ) 100. i 1 2 i i K i ∈ N | Now we have k = u+v√D (resp. k = (u+v√D)/2) if D 1(mod 4) (resp. i i 6≡ D 1(mod 4)). ≡ If v = 0 then D 100 (resp. D 400). Hence we just perform the − 6 | | ≤ | | ≤ algorithm for all imaginary quadratic fields up to this bound. For every (k ,k ) 1 2 ∈ A A we compute the unit equation k u + k u = 4 (see equation (3.2)). 1 2 1 1 2 2 × − For every solution of this, we make w = z d. Please note that up to k1,u1 k1,u1 now our arguments are independent of d. As we need z , in particular k1,u1 ∈ OK w . So with this condition we can forget about those elements where k1,u1 ∈ OK (w ) / Z. Next we factor (w )=r s2 with r Z/Z2 and (r,s)=1. NK k1,u1 ∈ NK k1,u1 · ∈ This way, we obtain the candidate shortlist d = s. Now we run the algorithm to compute (Q(√D)) within a finite set of pairs (d,D). For all these, we (1,1,1,d) AP get (Q(√ D))= (Q) iff (D,d)=( 1,1),( 1,2). (1,1,1,d) (1,1,1,d) AP − 6 AP − − If v =0 we have k 1,2,5,10 for all D. As we have previously dealt with i − ∈{ } the cases D 400 or D 100 (depending on D mod 4) we have to concern | | ≤ | | ≤ about D >400 or D >100, and then A = 1,2,5,10 and the units in the ring i | | | | { } of integers of Q(√ D) are just 1. The case now parallels the rational one and − ± (Q(√ D))= (Q). (1,1,1,d) (1,1,1,d) AP − AP 8 ENRIQUEGONZA´LEZ–JIMEŃEZANDJOSE´ M.TORNERO PleasenotethatSilverman[19,Theorem0.1]proves(amongmanyotherthings) thatif d 3,the only quadraticimaginaryfieldfor whichthere areMarkofftriples ≥ at all is D = 1. So we might have proved the theorem looking only at cases − (d,D) (1,D),(2,D),(d, 1) for any D and d 3. In any case, for these cases ∈ { − } ≥ we would still need the previous arguments. So, outstanding as Silverman’s result is, it also is of little use for the proof of this result. (cid:3) For the case of real quadratic case, we have put our algorithm to work in K = Q(√D),for d 103and1<∆ 104 (∆ beingthediscriminantof ). After K K K | |≤ ≤ O these computations, we can state the following conjecture. Conjecture 1. # Q(√D) =178. (1,1,1,d) (d,D)∈Z[/Z∗×Z/Z2AP (cid:16) (cid:17)   Moreover, the following table showes all the triples in a.p. over quadratic fields: d D Markoff triples ina.p. over Z[α]=O not inZ: first termand difference Q(√D) (α+2,α−3), (−α+2,α), (2α−1,−2α), (α+2,−α), (−4α+2,2α) −1 (−3α−4,α+3), (−2α−1,2α), (4α+2,−2α), (2,2α), (2,−2α) (3α−4,−α+3), (−α+2,−α−3) (4α+8,−4α), (11α−13,−23α−6), (−4α+8,4α), (−7α−7,7α) 2 (−11α−13,23α−6), (−35α−25,23α+6), (35α−25,−23α+6) (7α−7,−7α) 3 (9α+18,−9α), (−9α+18,9α) (−35α−5,22α+11), (22α−11,−22α−11), (−7α−6,4α+5) (−9α+8,22α+11), (35α+30,−22α−11), (−7α−9,2α+7), (α+4,−5α+2), (−3α+5,−2α−7), (−α+3,4α−1) 5 (7α+1,−4α+1), (−α+3,5α+7), (14α−3,−10α−2) 1 (−14α−17,10α+8), (9α+17,−22α−11), (−22α−33,22α+11) (3α+8,2α−5), (7α−2,−2α+5), (−9α+8,5α−2), (9α+17,−5α−7) (6α−1,−10α−8), (−6α−7,10α+2), (α+4,−4α−5) 6 (−6α−12,6α), (6α−12,−6α), (3α−3,−3α), (−3α−3,3α) 11 (−2α−4,4α−6), (6α−16,−4α+6), (−6α−16,4α+6), (2α−4,−4α−6) 14 (−2α−4,2α), (2α−4,−2α) 17 (4α+13,α−10), (−4α+9,−α−11), (6α−7,−α+10), (−6α−13,α+11) 21 (3α−3,9), (−3α+12,−9), (−3α−6,9), (3α+15,−9) (−11α−32,7α+14), (11α−21,−7α+7), (3α−4,−2α+5), 29 (−3α−7,2α+7), (α+7,−2α−7), (−3α−7,7α−7), (−α+6,2α−5), (3α−4,−7α−14) (−4α+15,α−10), (2α−3,−α+4), (5,−α−5), (2α−3,α+11) 41 (−2α−5,α+5), (5,α−4), (4α+19,−α−11), (−2α−5,−α+10) −1 (1,−α), (2α+1,−α), (1,α), (−2α+1,α) 2 (−2α+4,2α), (2α+4,−2α) 2 6 (3α−6,−3α), (−3α−6,3α) 11 (−3α−8,2α+3), (3α−8,−2α+3), (α−2,−2α−3), (−α−2,2α−3) 14 (−α−2,α), (α−2,−α) 3 (3α+6,−3α), (−3α+6,3α) 3 6 (α−1,−α), (−2α−4,2α), (−α−1,α), (2α−4,−2α) 21 (α+5,−3), (α−1,3), (−α+4,−3), (−α−2,3) 4 2 (α+2,−α), (−α+2,α) 6 6 (−α−2,α), (α−2,−α) 7 2 (α−1,−α), (−α−1,α) 9 3 (α+2,−α), (−α+2,α) 11 5 (−2α−3,2α+1), (2α−1,−2α−1) MARKOFF–ROSENBERGER TRIPLES IN ARITHMETIC PROGRESSION 9 In particular, if (d,D) Z/Z∗ Z/Z2 does not appear in the following tables, ∈ × then Q(√D) = (Q): (1,1,1,d) (1,1,1,d) AP (cid:16) (cid:17) AP d 1 D 1 2 3 5 6 11 14 17 21 29 41 − # (Q(√D)) 16 12 6 26 8 8 6 8 8 12 12 (1,1,1,d) AP d 2 3 4 6 7 9 11 D 1 2 6 11 14 3 6 21 2 6 2 3 5 − # (Q(√D)) 5 3 3 5 3 6 8 8 3 3 3 3 3 (1,1,1,d) AP Remark 2. Denote by the union of all quadratic fields, and for d Z define the 2 F ∈ plane curve :dY3 3Y2Z dX2Y 2X2Z =0. d D − − − Corvaja and Zannier [7, Corollary 1] tell us that ( ) is finite. The above Dd OF2 conjecture asserts that in fact we have been able to compute explicitely the set ( ). Moreover,it asserts that the uniparametric family , where d run over Dd OF2 Dd the rational intergers, has only a finite number of points over and gives all of OF2 them. After that, we have taken a longer step, and have computed many examples for (K) where K is any number field such that ∆ 104. In particular if (1,1,1,3) K AP | |≤ ∆ 104 then [K : Q] 7, after Minkowski’s bound. We have used the online K | | ≤ ≤ tablesofnumberfieldswithboundeddiscriminantfromthePARIgroup[14]. There are precisely 9115 such fields. In the table below we display the minimal polynomialfor a primitive element of K, along with the discriminant ∆ and the number n of Markoff triples in a.p. K K over (for those cases where there are more than the known rational triples), K O that is n =# (K). K (1,1,1,3) AP K ∆ n K ∆ n K K K K x2−3 12 6 x2−x−5 21 8 x2−6 24 8 x3−2 −108 6 x3−x2−8x−3 1425 6 x3−x2−6x+3 993 6 x3−x2−8x+9 1257 8 x3−x2−7x+4 1509 6 x3−x2−8x−1 1937 6 x3−9x−3 2673 6 x3−x2−13x+1 2292 6 x3−x2−9x+6 3021 6 x3−8x−2 1940 6 x3−9x−5 2241 6 x3−9x−2 2808 6 x3−10x−2 3892 6 x3−x2−10x+7 4065 6 x3−x2−15x−15 3540 6 x3−x2−18x+33 5073 6 x3−x2−21x+33 5172 6 x3−x2−13x−10 3877 6 x3−x2−12x+15 4281 5 x3−21x−24 5373 6 x3−12x−6 5940 6 x3−x2−23x+39 6108 6 x3−x2−12x+3 7473 6 x3−x2−17x+27 8628 6 x3−x2−18x+30 9192 6 x3−30x−27 9813 6 x4−x2+1 144 6 x4−x3−x2−2x+4 441 8 x4−2x2+4 576 8 x4+4x2+1 2304 6 x4+9 2304 8 x4+x2+4 3600 6 x4+6x2+18 4608 8 x4+3x2−6x+6 4752 6 x4+11x2+25 7056 8 10 ENRIQUEGONZA´LEZ–JIMEŃEZANDJOSE´ M.TORNERO x4−2x3−x2+2x+22 7056 6 x4−2x3+4x2+6 7488 6 x4−2x3−2x+1 −1728 6 x4−x3−3x2−x+1 −1323 8 x4−x3−x2−2x+1 −1791 5 x4−5 −2000 6 x4−x2−3x−2 −2151 5 x4−x3−x2−5x−5 −2475 6 x4−x3−x−2 −2943 6 x4−3x2−1 −2704 6 x4−4x2−3x+1 −2763 6 x4−2x2−4 −1600 6 x4−x2−3x+1 −3303 6 x4+x2−6x+1 −3312 6 x4−2x3−x2+2x−2 −3312 6 x4−2x3−2x+2 −3632 6 x4+3x2−9 −3600 10 x4−x3+2x2+x−2 −3951 6 x4−2x2−2 −4608 8 x4+2x2−2 −4608 6 x4−2x3+3x2+x−2 −4671 5 x4−2x3−3x−1 −4675 6 x4−3x2−6x−3 −5616 6 x4−2x3+3x2−2x−2 −5616 6 x4+2x2−11 −6336 8 x4−2x2−11 −6336 6 x4−x3+2x2−2x−2 −6444 6 x4−2x3+x2−3 −6768 6 x4−x2−6x−2 −6768 6 x4−3 −6912 6 x4−x3−4x−5 −6507 6 x4−2x2−3x+3 −6603 6 x4−x3−3x2+4x+2 −7668 6 x4−x3−x2+10x−20 −6975 7 x4−x2−3 −8112 6 x4−x3+x2+3x−3 −8739 5 x4−2x3−3x2+4x−2 −8640 8 x4−2x3−3x2−2x+1 −8640 8 x4−x3−x2−4x−2 −9012 6 x4−x3+5x2+x−2 −9036 6 x4−2x3−4x2−2x+1 −9408 6 x4+3x2−12 −9747 7 x4−x3−4x2+4x+1 1125 10 x4−6x2+4 1600 6 x4−2x3−7x2+8x+1 3600 6 x4−4x2+1 2304 24 x4−2x3−3x2+4x+1 4752 6 x4−2x3−4x2+5x+5 2525 8 x4−2x3−4x2+2x+1 7488 6 x4−5x2+1 7056 12 x4−x3−7x2+3x+9 4525 7 x4−2x3−7x2+2x+7 9792 6 x4−6x2−3x+3 9909 10 x4−x3−5x2+3x+4 8468 6 x4−5x2+2 9248 6 x5−2x4+x2−2x−1 −9759 6 The behaviour seems rather unpredictable here, including some instances where somefieldsdoappear,butnoneofitssubfieldsdo(liketheexamplewith∆ =1125 K above). And the existence of many examples close to the chosen bound prevents us to establish a conjecture for this case, as we did above. Remark 3. Here we present the results concerning the density of number fields of bounded discriminant which have non–rational Markoff triples in a.p.; let ∆ N ∈ and # K ∆ ∆, (Q)( (K) K (1,1,1,3) (1,1,1,3) r = || |≤ AP AP . ∆ (cid:8) # K ∆ ∆ (cid:9) K { || |≤ } Then ∆ 50 100 500 1000 5000 10000 r 0.088 0.042 0.015 0.009 0.012 0.0010 ∆ Data: All the Magma sources are available on the first author’s webpage. References 1. P. Alvanos, D. Poulakis: Solving genus zero Diophantine equations over number fields. J. SymbolicComput.46(2011)54–69.